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You can overlay it so your viewers can see it. You are basically adding an element to your video output. Which is pointless as they can see the chat room anyway.
You are not going to be able to see Twitch chat in game unless the game has it integrated.
I advise to get a second monitor or do what I do, view it on a tablet.

You -can- run your games in windowed mode and run the chat as always on top but unless you have a custom program to display IRC it's going to look like sh*t on top of cutting performance.